Sublevel Sets of the Wall-Confined Free Energy are Closed for the Wasserstein Distance

In the setting of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation, let $R$ be the wall radius of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §parameters and let $\mathcal{E}:\mathcal{D}\to\mathbb{R}$ be the wall-confined free energy of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §free-energy. $(\Sigma_{d,R},W_{2})$ is the complete metric space of The Noncommutative Laws with a Norm Bound Form a Complete Bounded Metric Space with Interpolation Points, and $\mathcal{D}\subseteq\Sigma_{d,R}$ by The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score §bounds.

1. (Closed sublevel sets)¶ Let $c$ be real, let $(\lambda_{m})_{m\in\mathbb{N}}$ be a sequence in $\mathcal{D}$ with $\mathcal{E}(\lambda_{m})\le c$ for every $m\in\mathbb{N}$, and let $\lambda\in\Sigma_{d,R}$ be such that the real sequence $(W_{2}(\lambda_{m},\lambda))_{m\in\mathbb{N}}$ converges to $0$. Then $\lambda\in\mathcal{D}$ and $\mathcal{E}(\lambda)\le c$.

2. (Lower semicontinuity)¶ $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to the metric space $(\Sigma_{d,R},W_{2})$.